No — but they're related. Icicles hanging off your gutters mean meltwater is reaching the cold eave, which is the same condition that creates ice dams. An ice dam itself is the ridge of ice that forms AT the eave and traps water behind it. You can have icicles without a dam, and a dam without obvious icicles.
Here's the tell: if you see thick ice building up along the eave line — not just icicles dangling from the gutter — you likely have an ice dam, and water is probably backing up behind it. Heavy icicles are still worth paying attention to, because they signal the melt-and-refreeze cycle is running. If you're not sure what you're looking at, send us a photo.
A few icicles after a sunny thaw are usually nothing. The ones to watch are large icicles fed straight from the gutter line, especially when they keep growing over several cold days — that means water is running to the eave and freezing there, the same engine that builds a dam. Pair that with a thick band of ice sitting on the eave and packed in the gutter, and you're almost certainly looking at a dam.
Check inside, too. Water stains on an upstairs ceiling or along an exterior wall, damp spots near the top of a window, or a gutter sagging away from the fascia are all signs a dam is already backing water up. The safest way to know is from the ground: take a photo of the eave from your driveway and send it to us at (330) 555-8301, and we'll tell you whether it's harmless or worth a same-day visit.